So recently I was studying delta epsilon method to prove limit. And I saw some easy limits to be proved formally using this method.
So I tried to frame some question of my own and I came across this limit
$$0(\displaystyle{\lim_{x \to \infty}} x)$$
My attempt:
Let x=$\frac 1 t$ then limit translates to :
$$0(\displaystyle{\lim_{t \to 0}} \frac 1 t)$$
which can be written as(or can it I have no idea) $$(\displaystyle{\lim_{t \to 0}} \frac 0 t)$$
After this I have no idea I cannot figure what is f(x)?(is it 0?? Doesn't seem right) or how to start with $ \left\lvert x-0 \right\rvert \lt \delta$ and arrive at a conclusion.
The limits seems to tend to zero graphically and intuitively to me but I cannot formalise.
Question:
How can we prove limit $$0(\displaystyle{\lim_{x \to \infty}} x)$$
exists or does not exists ?
(I am sorry if this a silly question to begin with)
Best Answer
The expression is undefined since $\lim\limits_{x\to\infty}x$ (which is computed before $0$ is introduced) does not exist, or is $+\infty$. See (mathcentre: Limits of functions pp. 3-4) and (Question 2347692). Your rearrangement using $a\lim\limits_{x\to c}f(x)=\lim\limits_{x\to c}af(x)$ does not hold because it is only valid when the limit in the LHS exists. However, you would be correct that $\lim\limits_{x\to \infty}0\cdot x=0$.